In this work, we propose a Deep Learning-based Reduced Order Model (DL-ROM) framework specifically crafted for addressing the peculiarities and challenges of physically parameterized differential problems featuring also geometrical variability and parametric domains. First, we focus on the properties of the available datasets, usually generated with synthetic solvers. Specifically, we observe that parameterized domain shapes often require different discretizations entailing a different number of degrees of freedom so that high-fidelity solvers can efficiently grasp the underlying physics. For this reason, we usually deal with multi-resolution datasets, and we claim that a continuous, infinite-dimensional framework is the most appropriate and versatile strategy to address problems featuring geometrical variability. Moreover, we emphasize that geometrical parameters generally have a profound impact on the solution manifold variability, whereas physical parameters usually have only a marginal effect. In this respect, we argue that an efficient surrogate model for geometrically parameterized problems should be made explicitly aware of the geometrical parameters, to better reflect the distinct influence that physical and geometrical parameters have on the solution variability. In this respect, we present Continuous Geometry-Aware DL-ROMs (CGA-DL-ROMs), along with a suitable abstract framework to set the dimensionality reduction task. Then, we demonstrate the effectiveness of our approach through a set of numerical experiments featuring both physical and geometrical parameterizations, ranging from simple test cases to industrial benchmarks.